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Evaluate 12000000000 times 10000000 over two hundredths times 20000000 times 500000, expressing it in scientific notation.
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I wanna show you two methods.
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The first one, we’ll simplify before we put this into scientific notation.
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And in the second method, we’ll put everything in scientific notation and then we’ll simplify.
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To simplify, what we’ll do is take the 12 and the one in the numerator and keep that.
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12 times one equals 12.
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And there are 16 zeros remaining.
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And that means we need a one followed by 16 zeros, like this.
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In the denominator, we have two times five.
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But that equals 10, which means it’s just another zero.
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For now, we won’t deal with this 0.02.
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We’ll just bring it down.
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And that is being multiplied by a one followed by 13 zeros.
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And we know that the 13 zeros on the bottom and the 13 zeros on the top can cancel out.
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We now have 12000 in the numerator and two hundredths in the denominator.
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To get the two hundredths out of the denominator, I wanna move the decimal two places to the right.
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And I can do that by multiplying by 100.
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But if I multiply the denominator by 100, I need to multiply the numerator by 100.
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And that means our numerator will be a 12 followed by five zeros, and our denominator will be two.
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We can then divide 12 by two, which equals six.
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And we’ll bring down our five zeros.
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This expression is now completely simplified.
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It needs to be put into scientific notation.
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Scientific notation requires that the ones place be between zero and nine.
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So we would rewrite this as six multiplied by 10 to some power.
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Well, we need to move the decimal five places to the right.
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And that means we’ll need 10 to the fifth power.
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This fraction in scientific notation is six times 10 to the fifth power.
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But what if we wanted to put everything in scientific notation before we multiplied and simplified?
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We’ll start with the same expression.
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And then we’ll write 12000000000 in scientific notation.
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That would be 1.2 times 10 to the 10th power.
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Next, 10,000,000 in scientific notation would be 1.0 times 10 to the seventh power.
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Moving into the denominator, how would we write two hundredths in scientific notation?
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2.0 times 10 to the negative two power.
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20000000 becomes 2.0 times 10 to the seventh power.
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And then 500000 is 5.0 times 10 to the fifth power.
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We notice that there is a 10 to the seventh in the numerator and the denominator.
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Those can cancel out.
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Our 10 to the fifth in the denominator will cancel, leaving 10 to the fifth in the numerator.
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Our numerator is now 1.2 times 10 to the fifth, and then our denominator two times 10 to the negative two times two times five.
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Two times five is 10, or you could write it as 10 to the first power.
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We can do some more cancelling.
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This 10 to the first power makes the numerator 10 to the fourth power.
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And then I wanna break these two pieces up.
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What is 1.2 divided by two?
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And then we’ll multiply 10 to the fourth over 10 to the negative two.
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1.2 divided by two or half of 1.2 is 0.6.
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And when we have a negative exponent in the denominator, it moves to the numerator.
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So we’ll have 10 to the fourth plus two power 10 to the sixth power.
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However, 0.6 is not in scientific notation.
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We can write 0.6 as six times 10 to the negative one power times 10 to the sixth power.
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Multiplying exponents with the same base means you add the values of the exponents.
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6.0 times 10 to the negative one plus six, 6.0 times 10 to the fifth.
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Both of these methods are completely accurate.
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And one of them just saved you from writing so many zeros.